3 Answers
3

From what I've read, there really is no deeper meaning other than that the outfits were dictated by the need to create contrasts between the major characters and the need to craft outfits that would be suitable for the various fight sequences.

This interview with Kym Barrett ("Costume Designer" for The Matrix 1, 2 and 3) indicates that the amount of input from the Directors was actually quite minimal. Simply put, you can't do kung-fu in tight leather pants.

KYM: "They always say, and they’ve never deviated from this, “We want it to be dark, we want it to be high contrast, we want Trinity to
be like an oil slick.” They give me simple sentences and let me go
away and come up with stuff, then we talk about it and it just grows.
After that we make [the garment], tweak it, and it grows into things.
Larry and Andy are great with allowing all of us (I think Owen
[Paterson, Production Designer] would agree), they give you a little
seed and then they let you expand it. Sometimes they think my ideas
are ludicrous and they’ll roll around the floor laughing, then they’ll
come back and say there was something there, even though it was a
ridiculous idea. It’s a good way to work; it’s a very challenging and
rewarding way to work

On one hand Larry and Andy are very broad, then on the other hand
they’re very detail oriented, so they work well seeing a body and
seeing how the light hits the body. The way the shape of the garment
moves in the air is really important to them, and you can’t show those
kind of things on a drawing. So, more often than not, they like to see
a prototype of something, rather than just a drawing."

And this interview with Gloria Bava ("Costume Cutter" for the Matrix 1, 2 and 3) shows that they wanted to clothe Neo in a non-shiny fabric to ensure an immediate visual contrast with Trinity and Morpheus. You should be able to tell their characters apart instantly.

GLORIA:Neo’s fabric was sourced around Sydney, and they found it in Sydney at a tailoring supply place. It had to be the right weight,
it had to be the right shade of black, it had to have a matte-ness
about it, not shiny in any way, and it had to have the right sort of
loose weave. It was more like a twill weave, which was very good to
work with, and it wasn’t a real black black, it was bordering on a
charcoal to black shade. When they found the fabric they bought lots
of it, and I think ended up making about four of those coats.

Neo's appearance in the Matrix notably shifts from the first movie to the rest of the trilogy. Whereas he takes a utilitarian approach in the first, coated in weaponry and tools, Reloaded and Revolutions show him in a coat which is very visually similar to the cassock - the garb of a priest.

This reflects the nature of his role as viewed by those who believe in the Prophecy; in their eyes, Neo is similar to a religious figure. The Prophecy as a religion is expressed throughout the films, perhaps most pointedly as a major point of contention between Cmdr. Lock(Deadbolt) and Morpheus:

Commander Lock: Dammit, Morpheus. Not everyone believes
what you believe.

Morpheus: My beliefs do not require them to.

Apart from the practicality of engaging in MMA-style fighting with leather clothes on (which "shouldn't matter" in the Matrix, but sadly our actors are subject to the physics of real life), it also gives his apparel a 'cape-like' effect to embellish his powers of flight. As it would happen, cassocks are not made of leather, they tend to be made of cotton and/or polyester.

At the end of the day it is another bit of real-world religious imagery that is injected into the movies to lend credence to Neo's status as the savior of the human race.

But he did have a leather jacket. I think that the clothing is more down to the individuals style as opposed to some rule that they have to wear leather. Because it'd be pretty easy for the machines to make a day where no one wears leather in the matrix thus the people who are free would stand out like sore thumbs.